Data.Octree.Internal:octantDistance from Octree-0.5.4.2

Average Error: 31.5 → 18.3

Time: 3.2s

Precision: binary64

Math

\[\sqrt{x \cdot x + y \cdot y}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -4.4451944394226077 \cdot 10^{121}:\\ \;\;\;\;-1 \cdot x\\ \mathbf{elif}\;x \le -1.61825031122827605 \cdot 10^{-247}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{elif}\;x \le -4.6382787383428388 \cdot 10^{-284}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \le 2.5717864824332804 \cdot 10^{-8}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

C

double code(double x, double y) {
	return ((double) sqrt(((double) (((double) (x * x)) + ((double) (y * y))))));
}

↓

double code(double x, double y) {
	double VAR;
	if ((x <= -4.445194439422608e+121)) {
		VAR = ((double) (-1.0 * x));
	} else {
		double VAR_1;
		if ((x <= -1.618250311228276e-247)) {
			VAR_1 = ((double) sqrt(((double) (((double) (x * x)) + ((double) (y * y))))));
		} else {
			double VAR_2;
			if ((x <= -4.638278738342839e-284)) {
				VAR_2 = y;
			} else {
				double VAR_3;
				if ((x <= 2.5717864824332804e-08)) {
					VAR_3 = ((double) sqrt(((double) (((double) (x * x)) + ((double) (y * y))))));
				} else {
					VAR_3 = x;
				}
				VAR_2 = VAR_3;
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Target

Original	31.5
Target	17.4
Herbie	18.3

\[\begin{array}{l} \mathbf{if}\;x \lt -1.123695082659983 \cdot 10^{145}:\\ \;\;\;\;-x\\ \mathbf{elif}\;x \lt 1.11655762118336204 \cdot 10^{93}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Derivation

1. Split input into 4 regimes

2. if x < -4.4451944394226077e121

   1. Initial program 55.7

      \[\sqrt{x \cdot x + y \cdot y}\]

   2. Taylor expanded around -inf 8.3

      \[\leadsto \color{blue}{-1 \cdot x}\]

   if -4.4451944394226077e121 < x < -1.61825031122827605e-247 or -4.6382787383428388e-284 < x < 2.5717864824332804e-8

   1. Initial program 21.3

      \[\sqrt{x \cdot x + y \cdot y}\]

   if -1.61825031122827605e-247 < x < -4.6382787383428388e-284

   1. Initial program 34.5

      \[\sqrt{x \cdot x + y \cdot y}\]

   2. Taylor expanded around 0 33.2

      \[\leadsto \color{blue}{y}\]

   if 2.5717864824332804e-8 < x

   1. Initial program 39.3

      \[\sqrt{x \cdot x + y \cdot y}\]

   2. Taylor expanded around inf 15.8

      \[\leadsto \color{blue}{x}\]
3. Recombined 4 regimes into one program.

4. Final simplification 18.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.4451944394226077 \cdot 10^{121}:\\ \;\;\;\;-1 \cdot x\\ \mathbf{elif}\;x \le -1.61825031122827605 \cdot 10^{-247}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{elif}\;x \le -4.6382787383428388 \cdot 10^{-284}:\\ \;\;\;\;y\\ \mathbf{elif}\;x \le 2.5717864824332804 \cdot 10^{-8}:\\ \;\;\;\;\sqrt{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array}\]

Reproduce

herbie shell --seed 2020163 
(FPCore (x y)
  :name "Data.Octree.Internal:octantDistance  from Octree-0.5.4.2"
  :precision binary64

  :herbie-target
  (if (< x -1.1236950826599826e+145) (neg x) (if (< x 1.116557621183362e+93) (sqrt (+ (* x x) (* y y))) x))

  (sqrt (+ (* x x) (* y y))))